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  • View in gallery

    Array of 18 disdrometers used for comparing modeling with actual measurements. The facility is located in Toledo City, Spain.

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    Pairwise scatterplots for several RDSD parameters for the four episodes analyzed in the paper: 4, 11, 14, and 23 Mar 2011.

  • View in gallery

    Difference in the Mie-calculated reflectivity factors of the Ku and Ka bands as a function of the m and σ. The calculations account for the finite integration range in D and the small but significant difference in attenuation between the echoes from the near end of the resolution volume and the far end.

  • View in gallery

    Validation of the T factor with empirical data calculated as the ratio between measured R and measured Z for the four episodes analyzed in the paper: 4, 11, 14, and 23 Mar 2011. Analytical T was calculated from (40) using the sample mean and variance of the RDSD.

  • View in gallery

    Analytical variability of the T factor in terms of the sample mean and variance as deduced from (40). Dots correspond to the disdrometer data of the four episodes used for validation: 4, 11, 14, and 23 Mar 2011.

  • View in gallery

    Convective–stratiform separation for some ZR relationships described in the literature in terms of (top) the T factor (R is in mm h−1) and (bottom) the a and b parameters in Z = aRb. The solid squares indicate stratiform (the S relations in the top graph) and black circles convective (C relations) rainfall. The star symbol and M-P stand for the Marshall and Palmer RDSD and the plus sign and WSR-88 for the Weather Surveillance Radar-1988 Doppler. The open circle indicates showers.

  • View in gallery

    (top to bottom) The four individual episodes: 4, 11, 14, and 23 Mar 2011. (left) Location of the empirical estimates in the Tmσ2 diagram. (middle) Time evolution of mσ2; the vertical coordinate represents the time evolution in 5-min aggregations. (right) Joint evolution of the mean (m), the variance (σ2), and the rain rate (R).

  • View in gallery

    (left) Main microphysical processes of rainfall in terms of the mean-variance diagram. As a precipitation system evolves, changes in m and σ2 translate into changes in the RDSD. (top to bottom) (middle) The eight major directions of change corresponding to precise microphysical processes and (right) faint lines in the RDSD plots representing the original RDSD; opaque red lines are the resulting RDSD. (Note that the area below both curves has to add to 1, as the RDSD represents probability densities in the probabilistic approach.

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A Probabilistic View on Raindrop Size Distribution Modeling: A Physical Interpretation of Rain Microphysics

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  • 1 Faculty of Environmental Sciences and Biochemistry, University of Castilla–La Mancha, Toledo, Spain
  • | 2 Jet Propulsion Laboratory, California Institute of Technology, Pasadena, California
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Abstract

The raindrop size distribution (RDSD) is defined as the relative frequency of raindrops per given diameter in a volume. This paper describes a mathematically consistent modeling of the RDSD drawing on probability theory. It is shown that this approach is simpler than the use of empirical fits and that it provides a more consistent procedure to estimate the rainfall rate (R) from reflectivity (Z) measurements without resorting to statistical regressions between both parameters. If the gamma distribution form is selected, the modeling expresses the integral parameters Z and R in terms of only the total number of drops per volume (NT), the sample mean [m = E(D)], and the sample variance [σ2 = E(mD)2] of the drop diameters (D) or, alternatively, in terms of NT, E(D), and E[log(D)]. Statistical analyses indicate that (NT, m) are independent, as are (NT, σ2). The ZR relationship that arises from this model is a linear R = T × Z expression (or Z = T−1R), with T a factor depending on m and σ2 only and thus independent of NT. The ZR so described is instantaneous, in contrast with the operational calculation of the RDSD in radar meteorology, where the ZR arises from a regression line over a usually large number of measurements. The probabilistic approach eliminates the need of intercept parameters N0 or , which are often used in statistical approaches but lack physical meaning. The modeling presented here preserves a well-defined and consistent set of units across all the equations, also taking into account the effects of RDSD truncation. It is also shown that the rain microphysical processes such as coalescence, breakup, or evaporation can then be easily described in terms of two parameters—the sample mean and the sample variance—and that each of those processes have a straightforward translation in changes of the instantaneous ZR relationship.

Corresponding author address: Francisco J. Tapiador, University of Castilla–La Mancha (UCLM), Faculty of Environmental Sciences and Biochemistry, Avda. Carlos III s/n, 45071 Toledo, Spain. E-mail: francisco.tapiador@uclm.es

Abstract

The raindrop size distribution (RDSD) is defined as the relative frequency of raindrops per given diameter in a volume. This paper describes a mathematically consistent modeling of the RDSD drawing on probability theory. It is shown that this approach is simpler than the use of empirical fits and that it provides a more consistent procedure to estimate the rainfall rate (R) from reflectivity (Z) measurements without resorting to statistical regressions between both parameters. If the gamma distribution form is selected, the modeling expresses the integral parameters Z and R in terms of only the total number of drops per volume (NT), the sample mean [m = E(D)], and the sample variance [σ2 = E(mD)2] of the drop diameters (D) or, alternatively, in terms of NT, E(D), and E[log(D)]. Statistical analyses indicate that (NT, m) are independent, as are (NT, σ2). The ZR relationship that arises from this model is a linear R = T × Z expression (or Z = T−1R), with T a factor depending on m and σ2 only and thus independent of NT. The ZR so described is instantaneous, in contrast with the operational calculation of the RDSD in radar meteorology, where the ZR arises from a regression line over a usually large number of measurements. The probabilistic approach eliminates the need of intercept parameters N0 or , which are often used in statistical approaches but lack physical meaning. The modeling presented here preserves a well-defined and consistent set of units across all the equations, also taking into account the effects of RDSD truncation. It is also shown that the rain microphysical processes such as coalescence, breakup, or evaporation can then be easily described in terms of two parameters—the sample mean and the sample variance—and that each of those processes have a straightforward translation in changes of the instantaneous ZR relationship.

Corresponding author address: Francisco J. Tapiador, University of Castilla–La Mancha (UCLM), Faculty of Environmental Sciences and Biochemistry, Avda. Carlos III s/n, 45071 Toledo, Spain. E-mail: francisco.tapiador@uclm.es

1. Introduction

Radar-derived precipitation is an important component of hydrological models. The uncertainty in the relationship between the radar reflectivity (Z) and the rain rate (R), or ZR relationship, is a major factor affecting precipitation estimates (Hossain et al. 2004) and thus hydrological model outputs. The study of the ZR relationship is therefore central to the hydrological applications of precipitation measurement from both ground and spaceborne radars such as Tropical Rainfall Measuring Mission (TRMM) and the forthcoming Global Precipitation Measurement (GPM) mission (Hou et al. 2008).

The derivation of the unknown variable (the R) from the radar measurement (the Z) has to be done either under a set of assumptions on the raindrop size distribution (RDSD) or relying on regression over ZR pairs. The later approach is the one used in radar operations. The expression for the ZR then takes the form of a power law: Z = a × Rb, with Z in mm6 m−3 and R in mm h−1. The choice of a power law dates back to the precomputer era and was motivated by the fact that it is easy to find by hand a and b simply by plotting log(Z) versus log(R) and fitting a straight line to the cloud of pairs; b and log(a) are then the slope and the intercept of the line, respectively. The operational method assumes that there is a collection of several (ZR) pairs measured over an extended period of time so that a regression can be performed. This is not a problem in practice since a disdrometer can sample at, say, a 5-min rate and provide simultaneous estimates of both parameters for the duration of a precipitation episode, for example, 2 h.

The trouble is that a and b lack physical meaning. While one can characterize different precipitation episodes and regimes in terms of different a and b (Chapon et al. 2008), these parameters are purely empirical and do not embed information on the rain microphysics. Moreover, the Z and the R varies continuously with time as a result of microphysical processes such as coalescence, breakup, or size sorting, so the regression, fitting line strategy implies averaging over instantaneous, qualitatively different RDSDs. In other words, apart from being unphysical, the a and b parameters from the regression represent some undefined averaged RDSD of the precipitation episode. Most important for remote sensing estimation of rain, they capture only the mean joint behavior. With the current availability of multiple simultaneous or nearly simultaneous measurements, all of which depend nonlinearly of the underlying rain, a mean relationship is not very useful and, when used by itself, will introduce bias in one’s combined retrievals (because a nonlinear function of a mean is not the mean of the nonlinear function).

To trace any physical roots on ZR relationships, one has to delve into the RDSD. Direct measurements of the RDSD are possible through disdrometers, as done by, for instance, Chandrasekar and Gori (1991), Bringi et al. (2003), Brawn and Upton (2008), and Cao and Zhang (2009), to name but a few. Since the RDSD measured by a disdrometer is always a sample of the whole population of drops, numerical methods have been devised to bridge the gap between sample statistics and population properties. Thus, sample data are used to calculate statistics such as the mean or the mode, and those are then used to derive the most likely analytical form of the RDSD for the whole population.

The many possible analytical forms of the RDSDs depend on several parameters that ideally relate in a transparent way to sample statistics. The goal of RDSD modeling is to relate parameters and statistics with the microphysical processes of rain, or, in other words, linking observables with parameters. Once this is done, the Z and R can be expressed as a function of variables accounting for the microphysics, and thus, a physically based ZR relationship can be built.

In the case of the gamma distribution modeling, which is the most widely used RDSD, there are three parameters—the slope, the shape, and the intercept of the distribution—determining the RDSD of the population. Much effort has been devoted to the task of establishing links between those particular parameters and observables. Problems, however, persist. The shape is only loosely related to the variance of the distribution as the location is with the mode, but the intercept parameter N0 (or ), which is defined as the maximum number of drops per volume at drop diameter D = 0, has no physical interpretation, not least because there are no zero-diameter drops.

Here it is shown that an alternative probabilistic modeling permits deriving a mathematically consistent RDSD from the number of drops per volume (NT), the sample mean [m = E(D)], and the sample variance [σ2 = E(mD)2] of the drop diameters (D). The resulting ZR depends on m and σ2 only, which provides a physical interpretation of a linear reflectivity–rainfall relationship. It is also shown that such formulation is equivalent to use NT, E(D), and E[log(D)] as the parameters to derive the RDSD. Such parameterization embeds a scaling law of the diameters. The main point of the paper in terms of precipitation remote sensing is that NT is not constant spatially or temporally and that it can be retrieved from ancillary information, such as a second frequency or passive measurements.

2. Empirical data

To compare the modeling with real data, RDSD measurements from a disdrometer array were used. The array is located in Toledo City, Spain, and consists of 18 Parsivel disdrometers (Loffler-Mang and Joss 2000) set within a few square meters for this experiment (Fig. 1). Sixteen of the instruments are of the Parsivel-1 type, whereas the other two are the more advanced Parsivel-2.

Fig. 1.
Fig. 1.

Array of 18 disdrometers used for comparing modeling with actual measurements. The facility is located in Toledo City, Spain.

Citation: Journal of Hydrometeorology 15, 1; 10.1175/JHM-D-13-033.1

Four rain episodes in March 2011 (Table 1) were sampled at 1-min intervals and then aggregated into 5-min accumulations in order to collect enough drops to ensure the representativeness of the samples. Only data consistent over the 16 identical disdrometers were used. Assuming Atlas and Ulbrich (1977) and Gunn and Kinzer (1949) relationships, drops with anomalous terminal velocity in relation to the diameter were filtered out. Thus, drops with a terminal velocity outside the υ(D) = 3.78D0.67 ± (3.78D0.67)/2 interval were eliminated. The dataset from a single disdrometer was used, but no significant difference was observed in the comparisons depending on the instrument choice.

Table 1.

Basic data of the four rain events.

Table 1.

3. Theory

a. Probabilistic versus statistical approach in RDSD modeling

In environmental modeling, there is a difference between a probabilistic and a statistical approach. Probability provides estimates of future outcomes based on a priori knowledge of the problem. Statistics, on the other hand, requires empirical observations to provide the estimate and thus is a posteriori knowledge.

In RDSD terms, the statistical approach is that represented by empirical fittings of analytical functions to drop measurements (Testud et al. 2001; Bringi et al. 2002; Kliche et al. 2008; Mallet and Barthes 2009; Kumar et al. 2010, 2011). The probabilistic approach, on the contrary, first provides a consistent mathematical framework to the problem, then provides a method to derive the parameters of the model, which are derived from sample statistics, and then validates the predictions with real data. The difference may look subtle, but it is critical to a deeper understanding of the ZR relationship and the microphysical processes behind it.

The idea behind the probabilistic modeling is to decouple the amount of rainfall from the shape of the distribution. A parameterization that could demonstrably decorrelate both would be very useful for rainfall estimation, specifically to perform single- or multiple-frequency retrievals. Issues such as parameter independence in the gamma distribution can only be fully explained with the probabilistic approach, as described below.

b. Probability distribution functions

The axiomatic approach to probability defines probability as a set of real numbers obeying a set of three properties. Thus, a probability distribution function (pdf) of raindrops of diameter D is a set of numbers p(D) ∈ ℝ, p(D) ≥ 0 satisfying
e1
where Ω is the population of the possible diameter values. Examples of pdfs are the normal, lognormal, gamma, beta, exponential, Poisson, or Weibull distributions, to name a few.

A pdf represents the properties of the whole population, but the parameters to define it are derived from the statistics of a sample. Thus, probability theory offers a method of bridging the gap between the properties of a small sample, such as disdrometer estimates of the precipitation crossing a small sample area, and the statistical properties of the population, such as the precipitation sensed by a radar beam over a large volume. In terms of RDSD modeling, a probability p(D) can be understood as a number that quantifies how likely is to find a drop of diameter D in a population of drops.

The choice of parameters for a particular form of the pdf satisfying (1) can be made in at least three different ways: using empirical fitting, through the classical formulation of probability distributions, and deriving the analytical form from the maximum entropy principle.

1) Empirical fitting

Empirical fitting was the path followed in early RDSD studies. It consists of setting up an a priori potential equation such as n(D) = acDb with a number of parameters. Then, the best match for the parameters and the sample data is found using minimization procedures such as the method of the moments or maximum likelihood. If the number of parameters is high enough, the equation is flexible enough to accommodate several situations, and the estimated RDSD compares well with empirical measurements. The major disadvantage of this approach is that the parameters are often correlated and that they do not have a clear physical interpretation, which hinders progress in the study of precipitation processes. Another problem is that the coefficients used for the fit may be linked to a particular region or regime and may perform poorly for other locations or episodes.

2) Classical probability theory

Classical probability theory establishes a more direct connection between the functional forms of the pdf and the features of the process being modeled. Thus, if a stochastic process occurs continuously and independently at a constant average rate, then it follows after some calculations that the probability of occurrence of a given outcome follows a Poisson distribution. The same applies to several others pdfs that are the result of several other stochastic processes. Thus, the binomial distribution accounts for the number of successes in a sequence of independent dichotomic and random trials, while the exponential distribution describes the time between events in a Poisson process. As described below, the gamma distribution can be also related with a definite stochastic process.

3) Maximum entropy formulation

The third method to select a pdf is maximum entropy. Jaynes (1957) described a method for selecting a pdf for a population using only the information contained in the sample with no further assumptions. Drawing on classical probability theory and on Shannon’s work (Shannon 1948), he found that any pdf satisfying (b) can be expressed as
e2
where λi are real numbers and gi are functions of D. These can be statistical or trigonometric moments, Fourier components, or any other function, including simple functions such as gi = D or gj = D2.
Expression (2) arises when the function
eq1
is maximized subject to the normalization property and a number of constraints. The general form of such constraints is
e3
Equation (2) is solved by finding the values of λ. If the number of constraints exceeds two, the problem is difficult to solve analytically, and numerical methods are required. In such cases, the method of Lagrangian multipliers is used and a numerical solution is found. For some particular instances, however, an analytical solution is possible. These solvable cases include the exponential, gamma, and lognormal distributions, which are those usually used in drop size distribution (DSD) modeling. Therefore, the classical and the maximum entropy formalisms are complementary. The former explains in terms of stochastic processes why a particular pdf arises and then gives the parameters to characterize it, whereas the latest provides a mean to find the least biased pdf of an unknown stochastic process from sample statistics. Thus, in the classical formulation, a gamma distribution of parameters μ and Λ models the waiting time for a drop of a given diameter to be found, providing the mean time of occurrence of that drop follows a Poisson process (Ross 1996; also see appendix). This process adequately models the measuring process of the RDSD and that helps explain why this particular pdf matches empirical data so well.

In terms of maximum entropy, the gamma distribution appears when the sample observables are the mean diameter and the mean logarithm diameter, as described below. No knowledge of the underlying stochastic process is needed in this case.

4. Modeling

a. Probabilistic interpretation of the gamma RDSD

The gamma distribution has been widely used in RDSD modeling because it provides a good fit with empirical data. It has, however, been used not as a pdf satisfying (1) but in the form of a number of empirical power-law expressions such as from Marshall and Palmer (1948) or Ulbrich and Atlas (1998):
e4
Attempts of providing more physically based formulation of the gamma distribution cause Testud et al. (2001) to formulate a normalized RDSD as
e5
where
e6
where Dm represents the mean particle size, is the intercept parameter of the exponential distribution of the same liquid water content and Dm, and μ is a parameter that describes the shape of the DSD. However, this solution is only partial as one can hardly agree that this description of and μ is physical at all, as claimed by the authors. Besides, the normalized approach does not account for maximum and minimum limits of the diameters. Neglecting those integration boundaries introduces a bias into integral parameters such as reflectivity and rain rate (Ulbrich 1985).

Other attempts to separate the NT and the pdf and to provide an analytical expression of the moments include the work of Chandrasekar and Bringi (1987), which noted that NT should be introduced into the RDSD modeling, and the normalization approach (Lee et al. 2004), which also aims to derive a universal pdf related to the local RDSD based upon the calculation of one or two moments.

The actual probabilistic expression for the gamma pdf satisfying (1) is
e7
where p(D) represents the probability of finding a drop of diameter D in the population of drops. Given this pdf, n(D), the number of drops per diameter D in a volume (the RDSD), can be calculated as
e8
where NT is the total number of drops per unit volume. This equation appears in the maximum entropy formalism as the result of selecting as sample functions:
e9
Thus, the constraints [see (3)] become
e10
To show this, it suffices to substitute the constraints into (2). This yields this pdf:
e11
which corresponds with the more familiar expression (7) if
e12
It can be shown that (11) is uniquely determined given those constraints and is not the result of coefficient identification. Once the gamma pdf is found, then the reflectivity (Z), the water content (W), and the rain rate (R) are derived as described in the following subsections.

b. Reflectivity (Z)

In the Rayleigh approximation, the reflectivity Z is related to the sixth power of the drop diameter D by
e13
If k1 = 1 and if D is in mm, then Z is in mm6 m−3. The integral can be analytically solved:
e14
The resulting expression for Z is
e15
If a maximum diameter is set, then the integral yields a slightly different expression:
e16
where γ (·) is the incomplete gamma function and Z becomes
e17
Further, if both a minimum and a maximum diameter are set, then from the properties of integral calculus it follows that
e18
and Z then becomes
e19

c. Water content (W)

The water content W relates to the third power of the drop diameter D:
e20
where if , then W is in cubic millimeters per cubic meter. As in section 4b, we obtain
e21

d. Rain rate (R)

The rain rate is by definition the flux of W, which implies modeling the velocity across a surface. The fall velocity is generally expressed as a power in the form
e22
typically with υ1 = 3.78 and υ2 = 0.67 (Ulbrich 1983). The units of υ1 are m s−1 mm−0.67. The rain rate is then calculated as
e23
where if , then R is in millimeters per hour. Operating as in (21) results in
e24

e. Generalized moment ratios

The analyses above can be used to derive the relationships between moments. Using the simplest form of the probabilistic RDSD,
e25
The M4,3, or mass weighted mean diameter Dm, is of particular importance in radar polarimetry. Applying the previous equation yields
e26
or, using the complete form with Dmin and Dmax,
e27
This result may be relevant for radar polarimetry as from the polarimetric measurement of reflectivity (Zh), differential reflectivity (Zdr) and specific differential phase (Kdp), estimates of the mass weight mean diameter Dm (or alternatively, the median diameter D0), the generalized intercept parameter Nw, and μ can be derived. This calculation can be done using, for instance, the method proposed by Gorgucci et al. (2002).

Using the modeling in the previous section, Dm shows its functional dependence with m and σ2, whereas appears as a function of m, σ2, and NT. As [see (28)], the three estimates needed to characterize the RDSD in the probabilistic approach can be derived from the polarimetric measurements. Since, as described below, m and σ2 appear related in real rainfall cases, only m and NT are required in practice.

5. Results

a. Independent parameters of the RDSD

In the equations for R and Z, there are only three parameters: m, σ2, and NT. There is no analytical way to recover any of the three parameters from the other two. The remaining variables are constants: k1 and k3 are values used to adjust for the units, whereas υ1 and υ2 are related with the modeling of the fall velocity of the hydrometeors. It is sensible that NT features in the modeling, as rain rate and the reflectivity must depend not only on the size distribution of drops (m) and on the spread of the diameters (σ2), but also on the actual number of drops.

All three parameters have a clear physical meaning in the probabilistic view of the problem. This contrasts with the use of N0, Λ, and μ for RDSD modeling. The intercept parameter N0 is no longer needed in the probabilistic formulation—a bonus of this modeling as this parameter has no physical interpretation (note that the units of N0 are physically implausible as m−4−μ). The normalized intercept parameter is not required either. The dependence between Λ and μ is made apparent when we consider that using the method of the moments in the probabilistic RDSD modeling yields (Thom 1958)
e28
where m and σ2 are not inherently correlated. In theory, m can take any value, and so can σ2, independently of the value taken by m. However once both are fixed, Λ is determined and then μ as a function of Λ. For instance, if one fixes μ = 0, then it follows that the RDSDs are constrained to those satisfying σ2 = m2.

Figure 2 shows the correlations between several RDSD-derived variables corresponding with four real precipitation episodes. As expected from theory, it is apparent that μ and Λ are correlated. The departure from the exact functional relationship μ = mΛ − 1 is due to the fact that real RDSDs are only approximately gamma. It is interesting to note that m and σ2 seem correlated for low values, presenting larger dispersion for higher values of both parameters. Individually, neither m nor σ2 are correlated with R or Z, while NT is with R.

Fig. 2.
Fig. 2.

Pairwise scatterplots for several RDSD parameters for the four episodes analyzed in the paper: 4, 11, 14, and 23 Mar 2011.

Citation: Journal of Hydrometeorology 15, 1; 10.1175/JHM-D-13-033.1

The independence of NT offers a mean to retrieve this value from radar and radiometer measurements. Using this modeling, the effective radar reflectivity factor Ze and the attenuation k can be calculated as (Meneghini et al. 2001)
e29
where σb and σe are the backscattered cross section and the extinction cross section at wavelength λ, respectively; kα are constants to account for the units, and |Kw|2 is the dielectric factor of water |Kw|2 = (j2 − 1)/(j2 + 2), with j as the complex refractive index of water. The variable σb,e can be computed using Mie theory.
For radiometer measurements, the integral scattering, extinction, and asymmetry parameters Ps, Pe, and Pa can be calculated as
e30
where a(D, λ) is the asymmetry factor. Using those equations, it is possible to retrieve NT and build tables for m and σ2.

The quantities m and σ2 are sensitive to the numbers of smaller drops, but that is only a problem if one were to try to estimate these two parameters by counting small drops using disdrometers. In the more interesting application of remotely sensing the microphysics using spaceborne radar, these parameters would be estimated from the measured radar reflectivity factors at two wavelengths (Ku and Ka bands in the case of GPM). Figure 3 displays the difference in the Mie-calculated reflectivity factors (accounting for the finite integration range in D and the small but significant difference in attenuation between the echoes from the near end of the resolution volume and the far end) as a function of the two parameters. Obviously, a single reflectivity difference cannot determine two unknown parameters, but one can use different assumptions about the joint behavior of the two parameters, that is, the correlation relation that is illustrated by three sample constant values for the coefficient of the relation, for example, to estimate the mean.

Fig. 3.
Fig. 3.

Difference in the Mie-calculated reflectivity factors of the Ku and Ka bands as a function of the m and σ. The calculations account for the finite integration range in D and the small but significant difference in attenuation between the echoes from the near end of the resolution volume and the far end.

Citation: Journal of Hydrometeorology 15, 1; 10.1175/JHM-D-13-033.1

From the figure, one can see, for example, that an observed large value for the reflectivity difference (green case), combined with the assumption that the coefficient of the σ2m1.5 relation is small (cyan line) allows one to deduce that m has a larger value, as opposed to the assumption that the coefficient of the σ2m1.5 relation is large (gray curve), which implies that m has a smaller value. The same applies to different possible values of the reflectivity difference: any measured value, combined with an assumption about the coefficient of the σ2m1.5 correlation, allows the estimation of m and σ2 without having to count (or be sensitive to) small drops.

b. Scaling law of drop diameters

As an alternative to (28), the maximum entropy approach yields (Singh et al. 1986)
e31
which corresponds with a maximum entropy formulation with two constraints. The variable Ψ is the digamma function defined as d[logΓ(·)]/d(·). This means that the appropriate form of the gamma RDSD can be found simply by estimating the arithmetic mean diameter E(D) and the logarithmic mean diameter E[log(D)].

Note that parameter estimation given in (31) arises when the constraints (10) are used. Both expressions have a definite interpretation: the first one reads that the diameters are monotonically growing values, and the second one indicates that there is a scaling law for those diameters. With these two assumptions, the gamma pdf follows from theory. If instead of (10), E[ln(D)] and var[ln(D)] are used as constraints, then the resulting pdf is the lognormal distribution.

Since this method is equivalent to (28), (31) provides an interesting interpretation of the gamma RDSD embedding a scaling law behavior of the diameters. The law relates the two constraints through the three Lagrangian multipliers. Thus, if a0, a1, and a2 are the three multipliers, then (Singh et al. 1986)
e32
and
e33
Therefore, an estimate of the mean value of the diameters and the value of the mean order of magnitude of the diameters would suffice to estimate a consistent gamma RDSD. This observation may be useful in remote sensing studies since the parameters are then related with the population expectations rather than with the sample means, as happens to be in the method of moments. Moreover,
e34
and
e35
which exposes the tight relationship between the covariance of D and log(D) in the probabilistic gamma RDSD modeling.

c. Z–R relationship

From (15) and (24), the functional relationship between Z and R, which is needed for retrieved rainfall rate from radar measurements, can be derived as
e36
which means that
e37
Noting
e38
then
e39
Here the relationship between Z and R appears naturally and not as the result of coefficient identification to a and b in the Z = aRb function, a method already dismissed in Haddad et al. (1997) as wrong.
If truncation is allowed and minimum and maximum diameters Dmin and Dmax are set, the expression for the T factor is
e40
Table 2 summarizes these results. It should be noted that in the probabilistic formulation the total number of drops per volume unit (NT) is required to calculate Z, W, or R, but not to derive T. The T in (40) or in its simplified version (39) is a function of m and σ2 only, both of which are physically meaningful, that is, they are quantities that we can measure with an instrument such as a disdrometer.
Table 2.

Summary of the RDSD modeling with the number of drops per volume NT, the sample mean m, and the sample variance σ2 as the independent variables, including the moments and the equations for the ZR relationship; Γ(·) is the gamma function and γ (·) is the incomplete gamma function.

Table 2.

Figure 4 shows that there is a good match between the analytical form of T as derived by theory (40) and the observed values. Differences can be due to departures from the diameter–fall speed relationship used, the imperfect gamma character of the distribution, and measurement and sampling errors.

Fig. 4.
Fig. 4.

Validation of the T factor with empirical data calculated as the ratio between measured R and measured Z for the four episodes analyzed in the paper: 4, 11, 14, and 23 Mar 2011. Analytical T was calculated from (40) using the sample mean and variance of the RDSD.

Citation: Journal of Hydrometeorology 15, 1; 10.1175/JHM-D-13-033.1

Figure 5 depicts the analytical variability of the T factor in terms of the sample mean and variance. The values of T span over several orders of magnitude. The figure gathers also the coordinates of real precipitation in this two-parameter diagram. The measurements of the March episodes lie in a narrow band of the configuration space. This is reasonable given the physics of the process: in real rainfall, one never observes RDSD as a few larger drops with no small or medium drops resulting from breakup or as many tiny drops that have not coalesced in their fall to yield a few large drops. These two cases would correspond with large m and small σ2 in the first instance and small m with large σ2 in the second. What we observe are RDSD where m and σ2 vary as the dots in Fig. 5, where individual 5-min RDSD have been plotted in the mean-variance parameter space. Apart from a few outliers that are probably due to experimental issues, it seems safe to state that the variance of the RDSD typically increases in a nonlinear way with the mean. For large variance, the dispersion of the mean seems to be large. This is consistent since large variance is associated with a few large drops, which are scarce in real rainfall and are subject to sampling problems when measured with disdrometers.

Fig. 5.
Fig. 5.

Analytical variability of the T factor in terms of the sample mean and variance as deduced from (40). Dots correspond to the disdrometer data of the four episodes used for validation: 4, 11, 14, and 23 Mar 2011.

Citation: Journal of Hydrometeorology 15, 1; 10.1175/JHM-D-13-033.1

d. Stratiform and convective rainfall partitioning

Because in real rainfall cases m and R are correlated and σ2 and R are also correlated, and since T depends on m and σ2, one cannot expect T and R to be independent. That is why the analytical linear relation (39) gives rise to nonlinear power-law ZR relations in practice for different rain events. Indeed, plotting R against T for several ZR relationships in the literature (Fig. 6, top) allows us to separate convective and stratiform rains. Compared with the a versus b of the Z = aRb empirical fit (Fig. 6, bottom), which has traditionally been used to that end (Ulbrich and Atlas 2002), the RT diagram has the advantage of providing a physical basis to the separation of both types of rain. The variables a and b lack a physical interpretation since they are just the coefficients in the log–log fit, but T is simply R/Z, so R versus T puts the convective–stratiform separation in terms of how large is the R/Z quotient at a given rainfall rate. In this diagram, the equilibrium RDSD (Z = 600R) corresponds with T = 1/600 = 0.0016 and features naturally as a straight line. This is consistent, as variations in the rainfall rate in statistically homogenous rain should depend on NT, not on T.

Fig. 6.
Fig. 6.

Convective–stratiform separation for some ZR relationships described in the literature in terms of (top) the T factor (R is in mm h−1) and (bottom) the a and b parameters in Z = aRb. The solid squares indicate stratiform (the S relations in the top graph) and black circles convective (C relations) rainfall. The star symbol and M-P stand for the Marshall and Palmer RDSD and the plus sign and WSR-88 for the Weather Surveillance Radar-1988 Doppler. The open circle indicates showers.

Citation: Journal of Hydrometeorology 15, 1; 10.1175/JHM-D-13-033.1

e. Microphysics in the mean-variance diagram

Figure 7 (left column) shows the location of 5-min disdrometer readings, disaggregated for the four selected episodes, in the mσ2 diagram. Within an overall similar pattern, the episodes present a distinct fingerprint on the diagram. By adding the time dimension to the plot (Fig. 7, middle column), the evolution of the rain episode can be traced in terms of the joint evolution of m and σ2, which are the only values determining the microphysics and thus the ZR relationship. Comparison with R (Fig. 7, right column) shows that the strong link between m, σ2, and R, which is nevertheless mediated by NT.

Fig. 7.
Fig. 7.

(top to bottom) The four individual episodes: 4, 11, 14, and 23 Mar 2011. (left) Location of the empirical estimates in the Tmσ2 diagram. (middle) Time evolution of mσ2; the vertical coordinate represents the time evolution in 5-min aggregations. (right) Joint evolution of the mean (m), the variance (σ2), and the rain rate (R).

Citation: Journal of Hydrometeorology 15, 1; 10.1175/JHM-D-13-033.1

The mσ2 diagram considerably simplifies relating changes in the RDSD with rain microphysical processes. Following Rosenfeld and Ulbrich (2003), the individual processes can be related with changes in the RDSD assuming everything else is held constant. While those authors elucidated the microphysics in terms of changes in the N(D) − D space, therefore integrating the number of drops into the modeling, here the discussion does not use NT. Even so, several microphysical processes can be traced.

As illustrated in Fig. 8, microphysics described by an m–σ2 pair can vary following eight main directions. For each of these directions there is a definite change in the RDSD that can be related with microphysical processes. Thus, if m decreases and σ2 increases (A in the figure) that means that there are both more small and large drops and that the number of small drops increases more than for the large drops. This corresponds with the physics of accretion, in which all drops across the spectrum grow by collecting cloud water. If both m and σ2 decrease (B), then the peak of the RDSD shifts to the left and the RDSD narrows. As a result, there are more small drops and less large drops. What has happened is that large drops have broken, generating many very small drops and a few medium-sized drops. This microphysical process is breakup. When m and σ2 both increase (C), the RDSD moves to the right and widens. This is equivalent to having less small drops and more large drops, something that happens if large drops collect small drops (coalescence). In the case of m increasing and σ2 decreasing (E), small drops have to decrease in order to maintain the same area under the curve since m has to move right, the RDSD has to narrow, and the area under both curves has to be same (and equal to one as the curves represent probabilities). The relative frequency of large drops, however, can either decrease or increase. In the first case, the process is the counterpart of the accretion (evaporation). In the second case, it is equivalent to coalescence (or size sorting). The equivalence of microphysical processes is not surprising and is found in other approaches to the problem. Thus, Rosenfeld and Ulbrich (2003) reported that the effect of updrafts on their NT-inclusive RDSD was undistinguishable from that of evaporation.

Fig. 8.
Fig. 8.

(left) Main microphysical processes of rainfall in terms of the mean-variance diagram. As a precipitation system evolves, changes in m and σ2 translate into changes in the RDSD. (top to bottom) (middle) The eight major directions of change corresponding to precise microphysical processes and (right) faint lines in the RDSD plots representing the original RDSD; opaque red lines are the resulting RDSD. (Note that the area below both curves has to add to 1, as the RDSD represents probability densities in the probabilistic approach.

Citation: Journal of Hydrometeorology 15, 1; 10.1175/JHM-D-13-033.1

If σ2 is held fixed (S1 and S2), then size permutation becomes the corresponding microphysical process. Depending on if m increases or decreases, we will have a process running in one or another direction; that is, the RDSD moving either to the left or the right. It is interesting to note that size permutation runs almost parallel to the T isolines (Fig. 7), meaning that the R/Z remains almost unchanged in the process. More complex processes such as up and downdrafts, or the case of a RDSD generated aloft and falling without any other process than sorting are more difficult to identify albeit any possible microphysics has to have a signature (probably not unique) into the diagram.

If m is the parameter that does not change, then if σ2 increases (A1) both small and large drops increase, which is an accretion process. The difference with the previous case above (A: decrease in m, increase in σ2) is that the change in the smallest drop has to be greater to compensate the widening of the RDSD while keeping the mean at the same time. In the case of σ2 decreasing (E1), then evaporation appears again. The difference with the case in which m increased (E) is that now the process takes place at both ends of the RDSD in a way that compensates and keeps the mean unchanged.

In practice, the observed correlation between σ2 and m (approximately σ2/m3 ~ 0.2 for our Toledo data) allows one to use the (NT, m, σ2) parameterization to make consistent simplifying microphysical hypotheses for dual-frequency-radar rain retrievals. Specifically, one can retrieve (NT, m, σ2) for each hypothesized value of chosen, for example, within two standard deviations of its mean. Indeed, the two radar reflectivity factors measured at two frequencies, in addition to the assumed value for , produce three equations that can be solved for the three unknowns. Using an empirical a priori distribution for , the retrievals can then either be averaged using this distribution or be further conditioned if an additional measurement is available (e.g., from a third radar channel). This approach accounts for the full nonlinearity in the ZR relations and is consistent with the empirical correlation between the microphysical parameters.

6. Summary and conclusions

Probability theory offers a mathematically consistent framework to model the RDSD. The advantages of the probabilistic gamma distribution modeling over empirical fittings are employing a consistent set of units, using three parameters with a clear physical meaning (m, σ2, and NT), and also that the modeling provides a ZR relationship that is independent on NT.

By decorrelating the number of drops NT from the shape of the distribution of diameters as in Chandrasekar and Bringi (1987), the rainfall amount can be treated as an independent parameter and is thus retrievable from satellite measurements. Since observed rainfall presents a high nonlinear correlation between m and σ2, only two parameters suffice. These parameters are sensitive to the number of smaller drops, but that is only a problem if one were to try to estimate them by counting small drops using disdrometers. In the case of remotely sensing the microphysics using spaceborne radar, these parameters can be estimated from the measured radar reflectivity factors at two wavelengths (such as Ku and Ka bands for GPM).

Within the probabilistic modeling of the RDSD, microphysical processes can be explained as changes in the m and σ2 parameter space. The a and b parameters of the power-law fit have been used to that end, but since they do not represent any physical feature of the RDSD, their values have not physical meaning by themselves. In contrast, m and σ2 have a clear meaning in terms of the functional form of the RDSD. Besides, a and b can only be found by statistical regression over several (Z, R) pairs and thus only capture the mean behavior or rainfall, whereas m and σ2 are instantaneous values.

Acknowledgments

Funding from projects PPII10-0162-5543 (JCCM), CGL2010-20787-C02-01, CGL2010-20787-C02-02 (MiCInn), Cenit project Prometeo (CDTI), CYTEMA, and UNCM08-1E-086 (MiCInn) is gratefully acknowledged. F.J.T. acknowledges Ramiro Checa and Luis Duran for double checking some early calculations and for their assistance in organizing and filtering the UCLM’s RDSD database. The authors also wish to acknowledge fruitful discussions in the Drop Size Distribution Working Group (DSDWG) of the Global Precipitation Measurement (GPM) mission.

APPENDIX

Poisson Processes and the Gamma Distribution

Following Ross’s formalism (Ross 1996), the measurements made by a disdrometer can been characterized as a Poisson process if it is assumed that 1) the number of drops in disjoint bins is independent and 2) the number of drops into any bin of size b is a random variable following a Poisson distribution of parameter λb. To prove that under these two assumptions the number of drops per given diameter follows a gamma distribution, we proceed as follows.

We are interested in estimating the pdf of the expected number of drops of diameter D. Having n drops, the probability of finding a drop in the interval [d, d + ε], with ε being a small interval, is the same as the probability of having exactly one drop in the interval [d, d + ε] and n − 1 drops in the interval [0, d) plus the probability of finding more than one drop in [d, d + ε]. Thus,
ea1
The probability of finding more than one drop in [d, d + ε] is negligible as ε is small. As the number of drops in disjoint intervals is independent and we have assumed a Poisson distribution, we have
ea2
Therefore,
ea3
Since n has to be an integer number (n − 1)! = Γ(n), we thus have a gamma pdf such as (7) with parameters Λ = λ and μ = n − 1. It follows that the Λ parameter in the RDSD is related with the rate parameter in the underlying Poisson distribution. A consequence of this modeling is that if Poissonian rain means steady rain, as suggested by Jameson and Kostinski (2002), then steady rain can be characterized as rain presenting a gamma RDSD.

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